Lyman-α forest and non-linear structure characterization in Fuzzy … · 2018. 9. 28. · Lyman-α...

MNRAS 000, 1–19 (2018) Preprint 28 September 2018 Compiled using MNRAS LATEX style file v3.0

Lyman-α forest and non-linear structure characterizationin Fuzzy Dark Matter cosmologies

Matteo Nori,1,2,3? Riccardo Murgia,4,6 Vid Iršič,5 Marco Baldi1,2,3 and Matteo Viel4,6,71Dipartimento di Fisica e Astronomia, Alma Mater Studiorum - University of Bologna, Via Piero Gobetti 93/2, 40129 Bologna BO, Italy2INAF - Osservatorio Astronomico di Bologna, Via Piero Gobetti 93/3, 40129 Bologna BO, Italy3INFN - Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna BO, Italy4SISSA, Via Bonomea 265, 34136 Trieste, Italy5University of Washington, Department of Astronomy, 3910 15th Ave NE, WA 98195-1580 Seattle, USA6INFN - Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Via Bonomea 265, 34136 Trieste, Italy7INAF - Osservatorio Astronomico di Trieste, via Tiepolo 11, I-34143 Trieste, Italy

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACTFuzzy Dark Matter (FDM) represents an alternative and intriguing description of thestandard Cold Dark Matter (CDM) fluid, able to explain the lack of direct detectionof dark matter particles in the GeV sector and to alleviate small scales tensions inthe cosmic large-scale structure formation. Cosmological simulations of FDM modelsin the literature were performed either with very expensive high-resolution grid-basedsimulations of individual haloes or through N-body simulations encompassing largercosmic volumes but resorting on significant approximations in the FDM non-lineardynamics to reduce their computational cost. With the use of the new N-body cos-mological hydrodynamical code AX-GADGET , we are now able not only to overcomesuch numerical problems, but also to combine a fully consistent treatment of FDMdynamics with the presence of gas particles and baryonic physical processes, in orderto quantify the FDM impact on specific astrophysical observables. In particular, inthis paper we perform and analyse several hydrodynamical simulations in order toconstrain the FDM mass by quantifying the impact of FDM on Lyman-α forest ob-servations, as obtained for the first time in the literature in a N-body setup withoutapproximating the FDM dynamics. We also study the statistical properties of haloes,exploiting the large available sample, to extract information on how FDM affects theabundance, the shape, and density profiles of dark matter haloes.

Key words: cosmology: theory – methods: numerical

1 INTRODUCTION

In the first half of the last century, the scientific commu-nity consensus gathered around two crucial facts about ourUniverse, that are now considered the pillars of modern cos-mology: firstly that the Universe is expanding and it is do-ing so at an accelerated rate and, secondly, that the esti-mated baryonic matter content within it cannot account forall the dynamical matter needed to explain its gravitationalbehaviour.

The standard cosmological framework built upon theseconcepts, called ΛCDM, still holds today. It implies the ex-istence of dark energy, as a source of energy for the accel-erated expansion of the Universe, and of dark matter, as anadditional gravitational source alongside standard matter,

? E-mail: [email protected]

without however specifying their fundamental nature thatstill represents a major puzzle for cosmologists.

The evidence for a cold and dark form of matter (CDM)– a not-strongly electromagnetically interacting particle ora gravitational quid that mirrors its effect – span over dif-ferent scales and are related to dynamical properties of sys-tems, as e.g. the inner dynamics of galaxy clusters (Zwicky1937; Clowe et al. 2006) and the rotation curves of spi-ral galaxies (Rubin et al. 1980; Bosma 1981; Persic et al.1996), but also to the gravitational impact on the under-lying geometry of space-time, as strong gravitational lens-ing of individual massive objects (Koopmans & Treu 2003)as well as the weak gravitational lensing arising from thelarge-scale matter distribution (Mateo 1998; Heymans et al.2013; Planck Collaboration et al. 2015; Hildebrandt et al.2017). Further evidence is based on the relative abundanceof matter with respect to the total cosmic energy budget

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required in order to reconcile Large Scale Structures (LSS)– as observed through low-redshift surveys – with the angu-lar power spectrum of CMB temperature anisotropies thatseed the early universe density perturbations (as observede.g. from WMAP and Planck Komatsu et al. 2011; PlanckCollaboration et al. 2016, respectively), on the clustering ofluminous galaxies (see e.g. Bel et al. 2014; Alam et al. 2017),on the abundance of massive clusters (Kashlinsky 1998) andtheir large-scale velocity field (Bahcall & Fan 1998).

Whether dark matter consists indeed of a yet unde-tected fundamental particle or it represents an indirect ef-fect of some modification of Einstein’s General Relativitytheory of gravity is still widely debated. Nevertheless, it hasbeen possible to exclude some of the proposed dark mat-ter effective models, such as e.g. the Modified NewtonianDynamics and its variants (MOND see e.g. Milgrom 1983;Sanders & McGaugh 2002; Bekenstein 2004), recently ruledout (Chesler & Loeb 2017) by the implications of the grav-itational wave event GW170817 (Abbott et al. 2017). Thelack of detection of dark matter particles in the GeV massrange through neither of indirect astronomical observations(see e.g. Albert et al. 2017), direct laboratory detections(see e.g. Danninger 2017), nor artificial production in high-energy collisions experiments (see e.g. Buonaura 2018) hasbeen undermining the appeal of the most massive dark mat-ter particle candidates, as e.g. the Weakly Interactive Mas-sive Particles (WIMPs), and it is presently shifting the sci-entific community efforts in the hunt of direct observationsfrom such high mass ranges towards lower ones (see e.g.Bertone et al. 2005).

A good starting point where to focus research and toclarify such long-standing uncertainties would be the appar-ent failures of the ΛCDM model at scales . 10 kpc – asgiven e.g. by the cusp-core problem (Oh et al. 2011), themissing satellite problem (Klypin et al. 1999), the too-big-to-fail problem (Boylan-Kolchin et al. 2012) –, all arising asan apparent inconsistency between simulations and obser-vations, the latter being more in line with less pronounceddensity fluctuations at those scales than predicted by theformer. However, the nature of such apparent failures hasbeen subject of debate in the astrophysics community. Itis still unclear, in fact, whether they should be ascribed toan imperfect baryonic physics implementation in numericalsimulations (see e.g. Maccio et al. 2012; Brooks et al. 2013),to an intrinsic diversity of properties related to the forma-tion history and local environment of each individual darkmatter halo (Oman et al. 2015), to the fundamental natureof the dark matter particle (see e.g. Spergel & Steinhardt2000; Rocha et al. 2013; Kaplinghat et al. 2000; Medvedev2014) or even to a combination of all these possible causes.

Among the particle candidates that have been proposedin the literature, Fuzzy Dark Matter (FDM) models describedark matter as made up of very light bosonic particles (seee.g. Hui et al. 2017, for a review on the topic), so light thattheir quantum nature becomes relevant also at cosmologicalscales. This requires a description of dark matter dynamicsin terms of the Schrödinger equation, in order to take intoaccount quantum corrections, and can be mapped in a fluid-like description where a quantum potential (QP) enters theclassical Navier–Stokes equation (Hu et al. 2000).

The typical wave-like quantum behaviour adds to thestandard CDM dynamics a repulsive effective interaction

that, along with creating oscillating interference patterns,actively smooths matter over-densities below a redshift-dependent scale that decreases with the cosmic evolution– as confirmed by FDM linear simulations (see e.g. Marsh &Ferreira 2010; Hlozek et al. 2015) – thus potentially easingsome of the previously mentioned small-scale inconsistenciesof the CDM model.

The lack of density perturbations at small scales in-duced by the QP is represented, in Fourier space, by a sharpsuppression of the matter power spectrum, that persists – atany given scale – until the action range of the QP shrinksbelow such scale and cannot balance any longer the effect ofthe gravitational potential (see e.g. Marsh 2016b, for anotherdetailed review on the subject). As a matter of fact, whilelinear theory predicts that perturbations at scales smallerthan the cut-off scale never catch up with those at largerscales – untouched by FDM peculiar dynamics –, non-linearcosmological simulations have shown that gravity is indeedable to restore intermediate scales to the unsuppressed level,in a sort of healing process (Marsh 2016a; Nori & Baldi2018).

FDM non-linear cosmological simulations have beenperformed over the years either with highly numerically in-tensive high-resolution Adaptive Mesh Refinement (AMR)algorithms able to solve the Schrödinger-Poisson equationsover a grid (see e.g. Schive et al. 2010, 2017) or with stan-dard N-Body codes that, however, include the (linear) sup-pression only in the initial conditions but neglect the inte-grated effect of the FDM interaction during the subsequentdynamical evolution (see e.g. Schive et al. 2016; Iršič et al.2017a; Armengaud et al. 2017) – basically treating FDM asstandard dark matter with a suppressed primordial powerspectrum, similarly to what is routinely done in Warm DarkMatter simulations (Bode et al. 2001). The former approachled to impressive results in terms of resolution (see e.g. Woo& Chiueh 2009; Schive et al. 2014) but is extremely com-putationally demanding, thereby hindering the possibilityof adding a full hydrodynamical description of gas and starformation for cosmologically representative simulation do-mains. On the other hand, the latter allows for such pos-sibility because of its reduced computational cost which is,however, gained at the price of the substantial approxima-tion of neglecting QP effects during the simulation (see e.g.Schive et al. 2016).

For these reasons, following the approach first proposedin Mocz & Succi (2015), we devised AX-GADGET (Nori &Baldi 2018), a modified version of the N-body hydrodynam-ical cosmological code P-GADGET3 (Springel 2005), to in-clude the dynamical effect of QP through Smoothed Parti-cle Hydrodynamics (SPH) numerical methods. The explicitapproximation of the dependence on neighbouring particlesresults in a less numerically demanding code with respect tofull-wave AMR solvers, without compromising cosmologicalresults, with the additional ability to exploit the gas and starphysics already implemented in P-GADGET3 , along with itsmore advanced and exotic beyond-ΛCDM extensions such asModified Gravity (Puchwein et al. 2013) or Coupled DarkEnergy models (Baldi et al. 2010).

Given that gravity, as mentioned above, can restore thesuppressed power at intermediate scales in the non-linearregime, major observables related to the LSS at such scalesmay appear similar in both FDM and CDM picture cosmolo-

MNRAS 000, 1–19 (2018)

Lyman-α and LSS properties in FDM cosmologies 3

gies at sufficiently low redshifts. For this reason, Lyman-αforest observations could play a crucial role in distinguishingsuch radically different models of dark matter, being one ofthe most far reaching direct astrophysical probes in terms ofredshift of the LSS observables, sampling the redshift rangez ∼ 2−5 (see e.g. Iršič et al. 2017a, for Lyman-α forest anal-ysis in N-body simulations, with neglected QP dynamicaleffects).

In this paper, we performed several simulations with themain goal of studying the effects of FDM on Lyman-α forestobservations in a fully consistent FDM set-up – i.e. withoutneglecting the QP during cosmic evolution – , in order toconstrain the FDM mass. As a by-product of our simula-tions, we are also able to perform an extended analysis ofthe statistical and structural properties of haloes, exploit-ing the large statistical sample at our disposal, to extractvaluable information about how FDM affects, among oth-ers, the halo mass function as well as the shape and densitydistribution of dark matter haloes.

The paper is organized as follows: in Section 2 we brieflydescribe the FDM models under consideration, providingall the basic equations that enter our numerical implemen-tation (2.1), and review the theoretical background behindLyman-α forest observations and its physical implications(2.2). In Section 3, we then recall how FDM dynamics isimplemented in the AX-GADGET code (3.1), we present thesimulation sets performed (3.2) and the strategy used toextract Lyman-α information (3.3); in Sections 3.4 and 3.5the procedures to deal with numerical fragmentation and tomatch haloes across different simulations are outlined. Theresults are collected in Section 4 and presented in decreasingorder of interested scale, including the matter power spec-trum (4.1), the Lyman-α statistics (3.3) and the structurecharacterization (4.3). Finally, in Section 5 we draw our con-clusions.

2 THEORY

In this Section we recall the main properties of a lightbosonic field in a cosmological framework, how it affects thegrowth of LSS and how Lyman-α forest analysis can be usedto probe these modifications.

2.1 Fuzzy Dark Matter models

The idea of describing dark matter and its key role in theLSS formation in terms of a ultra-light scalar particles – i.e.a particle with mass ∼ 10−22eV/c2 was introduced in Huet al. (2000), in which the term Fuzzy Dark Matter was usedfor the first time and the cosmological implications inducedby the quantum behaviour of such light dark matter field onlinear cosmological perturbations were outlined.

The Schrödinger equation describing the dynamics ofthe bosonic field φ̂i associated with a single particle can bewritten as

ih̄ ∂tφ̂i =−h̄2

m2χ∇2φ̂i+mχΦφ̂i (1)

where mχ is the typical mass associated with FDM particles– often represented in terms of m22 = mχ1022c2/eV – and

Φ is the gravitational potential, satisfying the usual Poissonequation

∇2Φ = 4πGa2ρb δ (2)

with δ = (ρ−ρb)/ρb being the density contrast with respectto the background field density ρb (Peebles 1980).

Under the assumption that all the particles belong to aBose-Einstein Condensate, the many-body field φ̂ of a col-lection of particles factorizes and the collective dynamicsfollows exactly Eq. 1. If this is the case, it is possible thento express the many-body field φ̂ in terms of collective fluidquantities as density ρ and velocity v, using the Madelungformulation (Madelung 1927)

v = h̄mχ=∇φ̂φ̂

(3)

ρ=mχ|φ̂|2 (4)

which translates into the usual continuity equation and amodified quantum Navier–Stokes equation reading

v̇ + (v ·∇)v =−∇Φ +∇Q (5)

where Q is the so-called Quantum Potential (QP)

Q= h̄2

2m2χ∇2√ρ√ρ

= h̄2

2m2χ

(∇2ρ2ρ −

|∇ρ|2

4ρ2

)(6)

also known as Quantum Pressure if expressed in the equiv-alent tensorial form as

∇Q= 1ρ∇PQ =

h̄2

2m2χ1ρ∇(ρ

4∇⊗∇ lnρ). (7)

The additional QP term accounts for the quantum behaviourof particles with a repulsive net effect that counteracts gravi-tational collapse below a certain scale, related to the Comp-ton wavelength λC = h̄/mχc identified by the boson mass(Hu et al. 2000). This can be heuristically viewed as theresult coming from two combined effects of quantum wave-like nature: decoherence, originating from the Heisenberguncertainty principle, stirring towards space-filling config-urations and interference creating oscillatory patterns (Huiet al. 2017).

In an expanding universe described by a scale factora and the derived Hubble parameter H = ȧ/a, the lineardensity perturbation δk in Fourier space satisfies – in thecomoving frame – the relation

δ̈k+ 2Hδ̇k+(

h̄2k4

4m2χa4− 4πGρb

a3

)δk = 0 (8)

that directly sets the typical scale

kQ(a) =(

16πGρba3m2χh̄2

)1/4a1/4 (9)

where the gravitational pull is balanced by the QP repulsion,sometimes referred as quantum Jeans scale in analogy withthe homonym classical one (Chavanis 2012).

The growing solution of Eq. 8, expressed in terms of thedimensionless variable x(k,a) =

√6 k2/k2Q(a), is

D+(x) =[(

3−x2)

cosx+ 3 xsinx]/x2 (10)

whose time dependence is bounded from above and below

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4 M. Nori et al.

by the large and small scale limits, respectively, as

D+ ∝{a for k� kQ(a)1 for k� kQ(a)

(11)

thereby recovering the standard ΛCDM perturbations evo-lution at large scales and halting growth of small scales over-densities (Marsh 2016b).

Structures are unable to collapse until the quantumJeans scale kQ(a) becomes so little that gravity can over-come the QP repulsive action and, in the linear perturba-tion regime, will forever carry information about their pastsuppressed state (Marsh 2016b). In non-linear simulations,instead, a recovery induced by gravity of the intermediatescales is indeed observed: in terms of matter power spec-trum, this implies that a portion of a FDM Universe, ob-served at a fixed scale, will eventually look like a CDM Uni-verse if a sufficient time for gravity recovery has passed afterthe crossing of the quantum Jeans scale (as argued also ine.g. Marsh 2016a).

All this considered, it is clear the reason why FDMmod-els peculiar imprints on LSS are to be looked for at verysmall scales for low redshifts, while larger scales may pro-vide relevant information only as long as higher redshifts areavailable to observations. In particular, FDM may reveal itspresence in the inner part of small collapsed haloes in theform of a flat solitonic core (see e.g. Schive et al. 2014; Marsh& Pop 2015; De Martino et al. 2018; Lin et al. 2018) whilelarger scales may show FDM imprints in the high-redshiftgas distribution (Iršič et al. 2017a; Armengaud et al. 2017;Kobayashi et al. 2017).

2.2 Lyman-α forest

The Lyman-α forest is the main manifestation of the in-tergalactic medium (IGM), the diffuse filamentary matterfilling the space between galaxies, and it constitutes a verypowerful method for constraining the properties of DM inthe small scale (0.5 Mpc/h . λ . 20 Mpc/h) and high red-shift regime (2 . z . 5) (see e.g. Viel et al. 2005, 2013).The physical observable for Lyman-α experiments is the fluxpower spectrum PF(k,z). Constraints on the matter powerspectrum from Lyman-α forest data at small cosmologicalscales are only limited by the thermal cut-off in the fluxpower spectrum, introduced by pressure and thermal mo-tions of baryons in the photo-ionised IGM. That is why thisastrophysical observable has provided some of the tightestconstraints up-to-date on DM scenarios featuring a small-scale power suppression (Iršič et al. 2017c; Murgia et al.2018), including FDM models, both in the case where theyconstitute the entire DM (Iršič et al. 2017a; Armengaudet al. 2017), and in the case in which they are a fractionof the total DM amount (Kobayashi et al. 2017).

Ultra-light scalar DM candidates are indeed expected tobehave differently with respect to standard CDM on scales ofthe order of their de Broglie wavelength, where they inducea suppression of the structure formation, due to their wave-like nature. In particular, for FDM particles with masses∼ 10−22eV , such suppression occurs on (sub)galactic scales,being thereby the ideal target for Lyman-α forest obser-vations. Moreover, as we discussed in the previous section,Lyman-α forest observations probe a redshift and scales

range in which the difference between ΛCDM and the FDMmodels – for the masses considered – is highly significant.

All the limits found in the literature on FDM param-eters – i.e. the mass mχ – using Lyman-α observations (ase.g. Iršič et al. 2017a; Armengaud et al. 2017; Kobayashiet al. 2017) have been computed by assuming that ultra-light scalars behave as standard pressure-less CDM and bycomparing Lyman-α data with flux power spectra obtainedfrom standard SPH cosmological simulations, which com-pletely neglected the QP effects during the non-linear struc-ture evolution. In other words, the non-standard nature ofthe dark matter candidate was simply encoded in the sup-pressed initial conditions used as inputs for performing thehydrodynamical simulations.

One of the goals of the present work is to use AX-GADGET in order to provide the first fully accurate con-straints on the FDMmass, by going beyond the standard dy-namical approximation of ignoring the time-integrated QPeffect. Including such effect in our numerical simulations isthereby expected to tighten the limits published so far inthe literature, since it introduces a repulsive effect at smallscales throughout the simulation evolution that contributesto the matter power spectrum suppression. Besides present-ing the new constraints, we will also carry out a meticulouscomparison with the bounds determined under the afore-mentioned approximation, in order to exactly quantify itsvalidity.

3 NUMERICAL METHODS

In this Section we briefly review the implementation of theAX-GADGET code routines that are devoted to the FDMdynamics (an in depth description featuring analytic andcosmological tests can be found in Nori & Baldi 2018). Wethen continue presenting how Lyman-α forest observationsare modelled and extracted from numerical simulations. Fi-nally, we describe our approach to discriminate spurioushaloes – which are expected to form in particle-based simula-tions featuring a suppressed power spectrum (see e.g. Wang& White 2007), such as Warm Dark Matter, Hot Dark Mat-ter, or FDMmodels – from genuine ones, in order to properlytake into account the known problem of numerical fragmen-tation, together with the strategy we used to cross-matchhaloes in the different simulations.

3.1 The code: AX-GADGET

AX-GADGET is a module available within the cosmologi-cal and hydrodynamical N-Body code P-GADGET3 , a non-public extension of the public GADGET2 code (Springel2005). It features a new type of particle in the system– i.e. ultra-light-axion (ULA) – whose strongly non-linearquantum dynamics is solved through advanced and refinedSmoothed Particle Hydrodynamics (SPH) routines, used toreconstruct the density field from the particle distributionand, therefore, to calculate the QP contribution to particleacceleration.

The general SPH approach relies on the concept thatthe density field ρ underlying a discrete set of particles canbe approximated at particle i position with the weighted

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sum of the mass m of neighbouring particles NN(i)

ρi =∑

j∈NN(i)

mjWij , (12)

where the mass is convolved with a kernel function Wij ofchoice, characterized by a particle-specific smoothing lengthhi, and whose extent is fixed imposing43πh

3i ρi =

∑j∈NN(i)

mj (13)

so that only a given mass is enclosed within it.Once the density field is reconstructed, every observable

is locally computed through weighted sums as

Oi =∑

j∈NN(i)

mjOjρjWij (14)

and its derivatives are iteratively obtained with

∇Oi =∑

j∈NN(i)

mjOjρj∇Wij (15)

where the derivation is applied on the window function.The exact scheme of the SPH algorithm is not fixed,

since each observable can be expressed in many analyticallyequivalent forms that, however, translate into different op-erative summations. For example, the QP of Eq. 6 can becalculated using recursive derivatives of ρ,√ρ or logρ in-termediate observables. An important consequence of suchflexibility is that different but analytically equivalent expres-sions will map into operative sums that carry different nu-merical errors. Among the several strategies that have beenemployed in the literature to reduce the residual numericalerrors (see e.g. Brookshaw 1985; Cleary & Monaghan 1999;Colin et al. 2006), the following has proven the more stableand accurate for the QP case (see Nori & Baldi 2018, fora comparison between different implementations), and willtherefore be the one of our choice:

∇ρi =∑

j∈NN(i)

mj∇Wijρj −ρi√ρiρj

(16)

∇2ρi =∑

j∈NN(i)

mj∇2Wijρj −ρi√ρiρj

− |∇ρi|2

ρi(17)

∇Qi =h̄2

2m2χ

∑j∈NN(i)

mjfjρj

∇Wij

(∇2ρj2ρj

−|∇ρj |2

4ρ2j

). (18)

AX-GADGET has undergone various stability tests andhas proven to be not only less numerically intensive with re-spect to Adaptive Mesh Refinement (AMR) full-wave solvers(Schive et al. 2010), due to the intrinsic SPH local approx-imation, but also to be accurate for cosmologically relevantscales as it agrees both with the linear (Hlozek et al. 2015)and the non-linear results (Woo & Chiueh 2009) available inthe literature, even if a proper convergence and code com-parison test has not yet been performed, since it would benecessary to assess the consistency of different numericalmethods at very small scales.

In fact, while cosmological and analytical results – ase.g. the soliton formation – are well recovered by N-Bodysimulations, interference patterns emerging at very small

scales seem more challenging to be represented accurately,due to their oscillatory nature that can be overly smoothedif the resolution – i.e. the number of particles – used is toolow. N-body simulations at very high-resolution – i.e. to thepc level – have yet to be performed, but as also argued inmore detail in Appendix A, it is our opinion that whetherinterference patterns can be observed or not is ultimately amatter of resolution.

The implementation of FDM physics in AX-GADGET includes the possibility to simulate Universeswith multiple CDM and FDM species or FDM particleswith self- or external interactions, as recently included withthe merging of the AX-GADGET module with the C-Gadgetmodule of Coupled Dark Matter models (Baldi et al. 2010).

Moreover, AX-GADGET inherits automatically all thelarge collection of physical implementations – ranging fromgas cooling and star formation routines to Dark Energy andModified Gravity implementations – that have been devel-oped for P-GADGET3 by a wide range of code developers.

All these properties allow to investigate a yet unex-plored wide variety of extended FDM models and make ofAX-GADGET a valuable tool – complementary to high res-olution AMR codes – to study the effects of FDM on LSSformation and evolution. In this work, we consider the sim-plest non-interacting case with the totality of the dark mat-ter fluid composed by FDM.

3.2 Simulations

In this work, we performed two sets of simulations, for a totalnumber of fourteen cosmological runs. The first set consistsin DM-only simulations used to characterize the small scalestructures at low redshift – i.e. down to z = 0 –, while thesecond one is evolved to z = 2 and includes gas particles anda simplified hydrodynamical treatment, as described in Sec-tion 3.1, specifically developed for Lyman-α forest analyses(the so-called "QLYA", or Quick-Lyman-alpha method, seeViel et al. 2004). Both sets consist in three pairs of sim-ulations, one pair for each considered FDM mass, evolvedeither including or neglecting the effect of the QP in the dy-namics – labelling these two cases as FDM and FDMnoQP,respectively –, in order to to assess and quantify the entityof such approximation often employed in the literature.

Both sets of simulations follow the evolution of 5123dark matter particles in a comoving periodic box with sidelength of 15Mpc, using 1Kpc as gravitational softening.The mass resolution for the dark-matter-only simulations is2.2124× 106M�. In all cases we generate initial conditionsat z = 99 using the 2LPTic code (Crocce et al. 2006), whichprovides initial conditions for cosmological simulations bydisplacing particles from a cubic Cartesian grid followinga second-order Lagrangian Perturbation Theory based ap-proach, according to a random realisation of the suppressedlinear power spectrum as calculated by axionCAMB (Hlozeket al. 2015) for the different FDM masses under investiga-tion. To ensure a coherent comparison between simulations,we used the same random phases to set up the initial con-ditions. In particular, the FDM masses mχ considered hereare 2.5×10−22, 5×10−22 and 2.5×10−21eV/c2, in order tosample the mass range preferred by the first Lyman-α con-straints in the literature (see in particular Iršič et al. 2017a;

MNRAS 000, 1–19 (2018)

6 M. Nori et al.

Armengaud et al. 2017; Kobayashi et al. 2017), obtainedthrough N-body simulations with approximated dynamics.

Cosmological parameter used are Ωm = 0.317, ΩΛ =0.683, Ωb = 0.0492 and H0 = 67.27 km/s/Mpc, As =2.20652× 10−9 and ns = 0.9645. A summary of the simu-lation specifications can be found in Tab. 1.

3.3 Lyman-α forest

The flux power spectrum PF (k,z) is affected both by astro-physical and cosmological parameters. It is therefore crucialto accurately quantify their impact in any investigation in-volving the flux power as a cosmological observable. To thisend, our analysis is based on a set of full hydrodynamicalsimulations which provide a reliable template of mock fluxpower spectra to be compared with observations.

For the variations of the mean Lyman-α forest flux,F̄ (z), we have explored models up to 20% different thanthe mean evolution given by Viel et al. (2013).

We have varied the thermal history of the Intergalac-tic Medium (IGM) in the form of the amplitude T0 andthe slope γ of its temperature-density relation, generallyparameterized as T = T0(1 + δ)γ−1, with δ being the IGMover-density (Hui & Gnedin 1997). We have then consid-ered a set of three different temperatures at mean density,T0(z = 4.2) = 7200,11000,14800 K, which evolve with red-shift, as well as a set of three values for the slope of thetemperature-density relation, γ(z = 4.2) = 1.0,1.3,1.5. Thereference thermal history has been chosen to be definedby T0(z = 4.2) = 11000 and γ(z = 4.2) = 1.5, providing agood fit to observations (Bolton et al. 2017). Following theconservative approach of Iršič et al. (2017a), we have mod-elled the redshift evolution of γ as a power law, such thatγ(z) = γA[(1 + z)/(1 + zp)]γ

S

, where the pivot redshift zpis the redshift at which most of the Lyman-α forest pixelsare coming from (i.e. zp = 4.2 for MIKE/HIRES+XQ-100).However, in order to be agnostic about the thermal historyevolution, we let the amplitude T0(z) free to vary in eachredshift bin, only forbidding differences greater than 5000 Kbetween adjacent bins (Iršič et al. 2017c).

Furthermore, we have also explored several values forthe cosmological parameters σ8, i.e. the normalisation of thematter power spectrum, and neff , namely the slope of thematter power spectrum at the scale of Lyman-α forest (0.009s/km), in order to account for the effect on the matter powerspectrum due to changes in its initial slope and amplitude(Seljak et al. 2006; McDonald et al. 2006; Arinyo-i Pratset al. 2015). We have therefore considered five different val-ues for σ8 (in the interval [0.754,0.904]) and neff (in therange [−2.3474,−2.2674]).

We have also varied the re-ionization redshift zrei, forwhich we have considered the three different values zrei =7,9,15, with zrei = 9 being the reference value and, finally,we have considered ultraviolet (UV) fluctuations of the ioniz-ing background, that may have non-negligible effects at highredshift. The amplitude of this phenomenon is parameter-ized by the parameter fUV: the corresponding template isbuilt from a set of three models with fUV = 0,0.5,1, wherefUV = 0 is associated with a spatially uniform UV back-ground.

Based on the aforementioned grid of simulations, we

have performed a linear interpolation between the gridpoints in such multidimensional parameter space, to obtainpredictions of flux power for the desired models.

We have to note that the thermal history implementa-tion of Iršič et al. (2017a) and the one used in this workare slightly different. For this reason, since the simulationsof the grid were performed without the introduction of theQP in the dynamics, we mapped our results into the gridones using the ratio between FDM and FDMnoQP simu-lations. This is, of course, not an exact procedure but weassume that the ratio of flux power spectrum with and with-out quantum pressure is relatively insensitive to the thermalhistory (Murgia et al. 2018).

In order to constrain the various parameters we haveused a data set given by the combination of intermediateand high resolution Lyman-α forest data from the XQ-100and the HIRES/MIKE samples of QSO spectra, respectively.The XQ-100 data are constituted by a sample of mediumresolution and intermediate signal-to-noise QSO spectra,obtained by the XQ-100 survey, with emission redshifts3.5≤ z ≤ 4.5 (López, S. et al. 2016). The spectral resolutionof the X-shooter spectrograph is 30− 50km/s, dependingon the wavelength. The flux power spectrum PF(k,z) hasbeen calculated for a total of 133 (k,z) data points in theranges z = 3,3.2,3.4,3.6,3.8,4,4.2 and 19 bins in k-space inthe range 0.003− 0.057s/km (see Iršič et al. 2017b, for amore detailed description). MIKE/HIRES data are insteadobtained with the HIRES/KECK and the MIKE/Magellanspectrographs, at redshift bins z = 4.2,4.6,5.0,5.4 and in 10k-bins in the interval 0.001−0.08s/km, with spectral resolu-tion of 13.6 and 6.7km/s, for HIRES and MIKE, respectively(Viel et al. 2013). As in the analyses by Viel et al. (2013) andIršič et al. (2017c), we have imposed a conservative cut onthe flux power spectra obtained from MIKE/HIRES data,and only the measurements with k > 0.005s/km have beenused, in order to avoid possible systematic uncertainties onlarge scales due to continuum fitting. Furthermore, we donot consider the highest redshift bin for MIKE data, forwhich the error bars on the flux power spectra are verylarge (see Viel et al. 2013, for more details). We have thusused a total of of 182 (k,z) data points. Parameter con-straints are finally obtained with a Monte Carlo MarkovChain (MCMC) sampler which samples the likelihood spaceuntil convergence is reached.

3.4 Numerical Fragmentation

For cosmological models whose LSS properties depart sensi-bly from ΛCDM only at small scales – as FDMmodels – , thethorough analysis of the statistical overall properties and thespecific inner structures of haloes represents the most rele-vant and often largely unexploited source of information.InN-body simulations, this implies the use of a suitable cluster-ing algorithm to build a halo catalogue in order to identifygravitationally bound structures that can then be studied intheir inner structural properties.

In this work, we rely on the SUBFIND routine alreadyimplemented in P-GADGET3 , a two step halo-finder whichcombines a Friends-Of-Friends (FoF) algorithm (Davis et al.1985) to find particle clusters – that defines the primarystructures of our halo sample – with an unbinding procedureto identify gravitationally bound substructures within the

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Table 1. Summary of the properties of the simulations set used for structure characterization.

Model QP in dynamics mχ [10−22eV/c2] N haloes N genuine haloes Mcut [1010M�]

LCDM × - 57666 56842 -

FDM-25 X 25 25051 13387 0.04056FDM-5 X 5 10058 2736 0.1645FDM-2.5 X 2.5 8504 1301 0.3151

FDMnoQP-25 × 25 25432 13571 0.04056FDMnoQP-5 × 5 10376 2856 0.1645FDMnoQP-2.5 × 2.5 8819 1374 0.3151

primary haloes (Springel et al. 2001). Hereafter, we use theterms primary structures to identify the substructures ofeach FoF group containing the most gravitationally boundparticle, subhaloes for the non-primary structures and haloeswhen we generally consider the whole collection of structuresfound.

However, a long-standing problem that affects N-bodysimulations, when characterized by a sharp and resolved cut-off of the matter power spectrum, has to be taken into ac-count in the process of building a reliable halo sample. Thisis the so-called numerical fragmentation, i.e. the formationof artificial small-mass spurious clumps within filaments (seee.g. Wang & White 2007; Schneider et al. 2012; Lovell et al.2014; Angulo et al. 2013; Schive et al. 2016).

While it has been initially debated whether the na-ture of such fragmentation was to be considered physical ornumerical, the detailed analysis by Wang & White (2007)showed that in Warm and Hot Dark Matter simulations (ase.g. Bode et al. 2001) – which are characterised by a highlysuppressed matter power spectrum – the formation of smallmass subhaloes was resolution dependent and related to thelarge difference between force resolution and mean parti-cle separation (as already suggested by Melott & Shandarin1989).

To identify spurious haloes in simulations and selecta clean sample to study and characterize the structures ofFDM haloes in each simulation, we take cue from the pro-cedure outlined in Lovell et al. (2014): in particular, we usethe mass at low redshift and the spatial distribution of par-ticles as traced back in the initial conditions as proxies forthe artificial nature of haloes as described below.

In fact, the more the initial power spectrum is sup-pressed at small scales, the more neighbouring particles arecoherently homogeneously distributed, thus facilitating theonset of artificially bounded and small ensembles that even-tually outnumber the physical ones. As already shown byWang & White (2007), the dimensionless power spectrumpeak scale kpeak and the resolution of the simulation – i.e.described through the mean inter-particle distance d – canbe related together to get the empirical estimateMlim = 10.1 ρb d / k2peak (19)describing the mass at which most of the haloes have a nu-merical rather than a physical origin. In Lovell et al. (2014),this mass is used as a pivotal value for the mass MCUTused to discriminate genuine and spurious haloes – lyingabove and below such threshold, respectively – which is setas MCUT = 0.5Mlim.

In addition to the mass discriminating criterion, Lovell

et al. (2014) showed that particles that generate spurioushaloes belong to degenerate regions in the initial conditionsand are more likely to lie within filaments, stating that thereconstructed shape of the halo particles ensemble in the ini-tial conditions can be used to identify spurious structures.N-Body initial conditions are generally designed as regularlydistributed particles on a grid from which are displaced inorder to match the desired initial power spectrum. Hence,numerical fragmentation originates mostly from particles ly-ing in small planar configurations, belonging to the samerow/column domain or a few adjacent ones.

Therefore, we need a method to quantitatively describethe shape of subhaloes and of the distribution of their mem-ber particles once traced back to the initial conditions of thesimulation. To this end, we resort to the inertia tensor of theparticle ensemble

Iij =∑

particles

m (êi · êj) |r|2− (r · êi) (r · êj) (20)

where m and r are the particle mass and position respec-tively and ê are the unit vectors of the reference orthonormalbase. The eigenvalues and the eigenvectors of the inertiatensor represent the square moduli and unitary vectors ofthe three axes of the equivalent triaxial ellipsoid with uni-form mass distribution. We define a ≥ b ≥ c the moduli ofthe three axes and the sphericity s = c/a as the ratio be-tween the minor and the major ones: a very low sphericitywill characterize the typical degenerate domains of numeri-cal fragmentation.

For these reasons, we use the combined information car-ried by the mass and the sphericity in the initial conditionto clean the halo catalogues from spurious ones by apply-ing independent cuts on both quantities as will be detailedbelow.

In Fig. 1 the mass-sphericity distributions of the differ-ent simulations are plotted at z = 0 (upper left panel) andat z = 99 (upper right panel) where each point representsa halo identified by SUBFIND, without applying any selec-tion. Solid and dash-dotted lines denote the median and the99th percentile of the distribution; in the side panels we dis-play the cumulative distributions, where the contribution ofspurious haloes is highlighted in black.

By looking at the two panels, it is possible to notice thatthe total cumulative sphericity distribution at low redshiftis fairly model independent, so that distinguishing spurioushaloes from genuine ones is impossible. However, if we tracethe particle ensembles of each halo found at z = 0 back tothe initial conditions at redshift z = 99, using particles ID,

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8 M. Nori et al.

z = 0 z = 99

mass [1010M ]

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LCDMFDM-25FDM-5FDM-2.5

Figure 1. Sphericities of all dark matter particles ensembles as found by SUBFIND as function of their mass (upper panels) at redshiftsz = 0 and z = 99 (left and right panels, respectively). The black-shaded area represents the discarded region below the different masscuts MCUT , corresponding to each model. Each black dot represents a subhalo and the solid (dot-dashed) lines describe the median(99th percentile) of the total distribution, which are all gathered and contrasted with ΛCDM in the lower panel. The total sphericitydistribution – integrated in mass – is represented in the side panels where the contribution of the discarded sample to medians anddistributions are portrayed in black. Lower panels feature the median of the mass-sphericity distributions, presented as the ratio withrespect to ΛCDM. The shaded areas, corresponding to the ±1σ of the distribution, are colour-coded as in the upper panels. The blackenedmedian and shaded areas represent the excluded portion of the sphericity distributions below the corresponding MCUT .

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and we study the resulting reconstructed mass-sphericity re-lation, the anomalous component of the distribution associ-ated with spurious haloes clearly emerges as a low-sphericitypeak, which is more pronounced for smaller values of theFDM particle mass.

In fact, as the mass mχ decreases, the smoothing ac-tion of the QP becomes more efficient, inducing homogeneityat larger and larger scales in the initial conditions and in-creasing, consequently, the contamination of numerical frag-mentation. It clearly appears that the population of haloesin the initial conditions is homogeneously distributed inΛCDM while a bimodal structure emerges at lower andlower FDM mass. In particular, an increasing number ofhaloes are located in a small region characterized by lowmass (M . 109M�) and low sphericity (s. 0.20).

As there is no theoretical reason why the QP shouldfavour the collapse of ensembles with very low sphericitiesin the initial conditions with respect to the ΛCDM case, weconsider this second population as the result of numericalfragmentation.

As in Lovell et al. (2014), we choose to computeMCUT = 0.5Mlim using Eq. 19 – one MCUT for each valueof the FDM mass, as reported in Tab. 1 –, that define theupper bound of the discarded mass regions, i.e. the black-shaded areas in all panels of Fig. 1.

It is interesting to notice that the massesMCUT appearto be very close to the values at which the sphericity mediansof the simulation sample – in the initial conditions – departfrom the ones of ΛCDM, as can be seen in the lower-rightpanel of Fig. 1. As the MCUT values we obtain are slightlylarger compared to these departing values, we confirm thechoice of the former over the latter, as a more conservativeoption for the mass thresholds dividing spurious from gen-uine haloes.

In Lovell et al. (2014), the selection in terms of initialsphericity was operationally performed discarding every halowith sphericity lower than sCUT = 0.16, equal to the 99thpercentile of the distribution of haloes with more than 100particles in the ΛCDM simulation. In our set of simulations,a similar value denotes the 99th percentile as measured atthe MCUT mass in each simulations, so we adopt it as ourown threshold in sphericity. Let us stress that the haloesthat are discarded through sphericity selection in the initialconditions have sphericities at z = 0 that are statisticallyconsistent with the genuine sample, making their numericalorigin impossible to notice based only on the sphericity dis-tribution at z = 0. However, the mass constraint is far morerigid than the sphericity one in all models but ΛCDM, whereno mass limit is imposed.

Finally, in Tab. 2 we have summarized the comparisonof the number of haloes in the FDMnoQP set-up with re-spect to the corresponding FDM set-up, presented as theratio of the total number of haloes found by SUBFIND andthe number of genuine haloes remaining after the exclusionof spurious ones. It is possible to see that in the FDMnoQPsimulations, for the three FDM masses considered, the totalnumber of haloes is overestimated by a factor ∼ 2.5% on av-erage while the genuine haloes excess becomes more impor-tant as the FDM mass decreases, up to 5.6% for m22 = 2.5.This means that neglecting the effects of the QP during thesimulation leads to the formation of haloes which are notpresent when the full QP dynamics is taken into account

Table 2. The total and genuine number of haloes, presented asthe ratio between the simulations neglecting and considering theQP dynamical effects.

mχ [10−22eV/c2] N haloes N genuine haloes

25 101.6% 101.4%5 103.5% 104.4%2.5 103.1% 105.6%

and that, using our à la Lovell et al. (2014) spurious detec-tion selection, such haloes pass the numerical fragmentationtest and contaminate any halo statistical property charac-terization.

3.5 Inter-simulations halo matching

In FDM models, as we said in the previous sections, notonly the initial power spectrum of matter perturbation issuppressed at small scales, thereby preventing the forma-tion of small mass structures, but the dynamical evolutionof density perturbations changes due to the effect of theQP, intimately affecting the development of structures dur-ing the whole cosmological evolution by opposing gravita-tional collapse. The implementation of such effect in AX-GADGET breaks the one-to-one correspondence of the spa-tial position of collapsed structures in simulations with dif-ferent FDM masses – especially for smaller objects –, despitethe identical random phases used to set up the initial con-ditions.

We indeed expect bigger haloes not to change dramat-ically their position at low redshift across different simula-tion, while this is not the case for lighter subhaloes whichare more affected by the evolving local non-linear balancebetween gravity and the QP of the environment.

This makes it more difficult to identify matching col-lapsed objects of common origin across the simulations, tostudy how FDM models affect the inner structure of haloeson a halo-to-halo basis.

To this end, we devise an iterative matching procedure,to be repeated until no more couples are found, as the fol-lowing: given a halo i at position ri and total mass mi insimulation A,

(i) select all haloes j belonging to simulation B as po-tential counterparts if |ri− rj |/(ai+aj) < R̃ where ai andaj are the major axes of the haloes computed through theinertia tensor of all their member particles.(ii) within the ensemble selected at the previous point, re-

tain only the haloes k⊆ j whose masses satisfy the condition|mi−mk|/(mi+mk)< M̃(iii) if more than one halo l ⊆ k is left, then choose the

one for which |ri−rl|/(ai+al) is minimum.(iv) after having considered all the haloes in A, if more

than one are linked to the same halo l belonging to B,choose the couple (i, l) that minimizes [|ri−rl|/(ai+al)]2 +[|mi−ml|/(mi+ml)]2, in order give the same weight to thetwo criteria.

This method is flexible enough to account for the shiftin mass and position we expect from simulations with differ-ent FDM mass models, but conservative enough to ensure

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10 M. Nori et al.

Table 3. Number of common matches across LCDM and FDMssimulations, using different values of the parameter M̃ represent-ing the minimum allowed ratio between the minimum and maxi-mum masses of each candidate couple.

M̃ mmin/mmax N matches

1/39 95% 531/19 90% 1623/37 85% 2341/9 80% 2791/7 75% 3041/3 50% 3463/5 25% 3611 0% 389

the common origin of the subhalo couples. Moreover, usingthe combination of position and mass filters, we are able todiscriminate couples in all mass ranges: position filtering isweaker constraint in the case of bigger haloes – since theyoccupy a big portion of a simulation – where instead themass filter is very strict; vice-versa, it is more powerful forsmaller haloes for which the mass filter select a large numberof candidates.

Operatively, we use the previous procedure to matchhaloes in each simulation with the ΛCDM one and we referto the subset of haloes that share the same ΛCDM compan-ion across all the simulations as the common sample.

For geometrical reasons, we set the limit value for R̃to be 0.5: this represents the case in which two haloes withthe same major axis a have centres separated exactly bythe same amount a. The configurations that are selected bypoint (i) are the ones for which the distance between the halocentres is less or equal the smallest major axis between thetwo. A higher value for R̃ would include genuine small haloesthat have been more subject to dynamical QP drifting butwould also result in a spurious match of bigger haloes. Forthese reasons, we adopt R̃ = 0.5, checking that the selectedsample gains or loses ∼ 5% of components if values 0.45and 0.55 are used, without modifying the overall statisticalproperties of the sample itself.

With respect to M̃ at point (ii), instead, we appliedthe matching algorithm using several values, each denotinga specific threshold of the minimal value allowed for the massratio of halo couples in order to be consider as a match. Asreported in Tab. 3, more than 60% of all the matching haloesacross LCDM and FDMs simulations without mass selection– M̃ = 1 case – have a mass ratio in the 100−85% ratio rangeand almost 80% in the 100−75% range. In order not to spoilour matching catalogue, especially with very close but highlydifferent in mass halo couples, we choose the limiting valueof M̃ = 1/7.

4 RESULTS

In this Section we present the results obtained from oursimulations in decreasing order of scale involved, startingfrom the matter power spectrum, to the simulated Lyman-α forest observations, to the statistical characterization ofhalo properties and their density profiles.

4.1 Matter Power spectrum

The relative difference of the matter power spectrum of thevarious FDM models with respect to ΛCDM at four differentredshifts is displayed in Fig. 2.

As already found in the literature (see e.g. Marsh 2016a;Nori & Baldi 2018), the evolution of the matter power spec-trum shows that the initial suppression – encoded in thetransfer functions used to build up the initial conditions –is restored at intermediate scales to the unsuppressed level,eventually, by the non-linear gravitational evolution.

At the redshifts and scales that are relevant for Lyman-α forest observations, however, the relative suppression withrespect to ΛCDM is still important and ranges from 5−20%for the lowest FDM mass considered.

The relative difference of the matter power spectrum,displayed in Fig. 3, shows an additional suppression withrespect ΛCDM (by a factor ≈ 1.15) when the QP is includedin the dynamical evolution (i.e. in the comparison betweenthe FDM and the FDMnoQP simulations). This is consistentwith the QP full dynamical treatment contributing as anintegrated smoothing force that contrasts the gravitationalcollapse of the otherwise purely collisionless dynamics.

4.2 Lyman-α forest flux statistics

In order to build our simulated Lyman-α observations we ex-tracted 5000 mock forest spectra from random line-of-sightswithin the simulated volume. The spectra are extracted ac-cording to SPH interpolation and the ingredients necessaryto build up the transmitted flux are the HI-weighted pecu-liar velocity, temperature and neutral fraction. Among thedifferent flux statistics that can be considered, we focus onthe flux probability distribution function (PDF) and fluxpower spectrum. Unless otherwise stated we normalize theextracted flux arrays in order to have the same observedmean flux over the whole sample considered and for all thesimulations. In any case, we do find that the scaling factorfor the optical depth arrays over the whole simulated volumeis 1.6, 1.4 and 1.1 times higher than in the ΛCDM case inorder to achieve the same mean flux for them22 = 2.5, 5 and25 FDM cases with negligible – between 1−2% – differencesbetween the FDMs and FDMnoQP cases.

In Fig. 4 we show the flux (top panel) and gas (bottompanel) PDF ratios between the simulations that include theQP and those that do not include it – FDM and FDMnoQP,respectively– at z = 5.4, one of the highest redshift bins inwhich Lyman-α data are available.

It is possible to see that there is a 2−6% peak at flux∼ 0.6−0.8, i.e. in regions of low transmissivity that are ex-pected to trace voids. The fact that FDM simulations dis-play a more peaked PDF compared to FDMnoQP ones forthis range of fluxes means that, on average, in those modelsit is more likely to sample such void environments. In fact,the different PDFs should reflect the underlying different gasPDFs at the same redshifts and along the same lines-of-sight.In the bottom panel of Fig. 4, showing the corresponding gasPDF, it is indeed apparent that in models with FDMs thegas PDF is more skewed towards less dense regions, that aretypically associated to high transmission. The effect due tothe QP is thus to increase the volume filling factor of regions

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z = 9 z = 5.4

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Figure 2. Matter power spectrum of FDM models contrasted with LCDM at different redshifts.

below the mean density with respect to the correspondingFDMnoQP case.

In Fig. 5 we plot the percentage difference in terms offlux power spectrum at three different redshifts and for theFDM models, both compared to ΛCDM (right panels) andto the corresponding FDMnoQP case (left panels). The in-crease of power at z = 5.4 in the largest scales – comparedto the ΛCDM case – is due to the imposed normalization atthe same mean flux, while the evident suppression at smallscales is related to the lack of structures at those scales. Thecomparison with the FDMnoQP set-ups, instead, reveals anadditional suppression which is always below the 5% levelfor all the masses considered. Since the flux power spectrumis an exponentially suppressed proxy of the underlying den-sity field, these results are consistent with the matter powerspectrum results previously shown in Fig. 2 and Fig. 3.

Since the Lyman-α constraint are calculated by weight-ing the contribution from all the scales, we expect the boundon m22 found in Iršič et al. (2017a) to change comparablyto the additional suppression introduced, that in our case is2−3%.

This is exactly what can be seen in in Fig. 6, wherethe marginalised posterior distribution of mχ obtained in

the present work is plotted and compared with the resultspresented in Iršič et al. (2017a). The red line refers to ourMCMC analysis, whereas the green line corresponds to theresults obtained by Iršič et al. (2017a). The correspondingvertical lines show the 2σ bounds on the FDM mass. The2σ bound on the FDM mass changes from 20.45×10−22 eVto 21.08× 10−22 eV, which matches with our expectationand confirm that the approximation of neglecting the QPdynamical effects in Iršič et al. (2017a) was legitimate toinvestigate the Lyman-α typical scales. The agreement be-tween the sets of results obtained with and without the dy-namical QP implementation is evident and is not sensiblyaffected by varying the assumptions on the IGM thermalhistory.

This result represents – to our knowledge – the firstFDM mass constraint derived from Lyman-α forest obser-vations that accounts for the full non-linear treatment ofthe QP, which introduces an additional – albeit not big –suppression of the matter power spectrum in the redshiftrange and comoving scales probed by the Lyman-α forest.The agreement with previous results implies that the non-linear evolution of the large-scale structure and the non-

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12 M. Nori et al.

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Figure 3. Matter power spectrum percentage differences between FDM simulation and their FDMnoQP counterpart at different redshifts.The difference in power spectrum suppression of having the QP in the dynamics result in a multiplication of ∼ 115% factor of thesuppression with respect to LCDM of Fig. 2 at scales k ∼ 10 hMpc−1.

linear mapping between flux and density effectively makeup for the additional suppression introduced.

4.3 Structure characterization

The statistical properties of the genuine haloes belonging toeach simulation are summarised in Fig. 7, where we displaythe cumulative halo mass function (top right panel), the halomass outside R200 (top left panel) – where R200 identifiesthe distance from the halo centre where the density is 200times the critical density of the Universe andM200 the masscontained within a R200 radius sphere –, the subhalo massfunction (bottom left panel), and the subhalo radial distri-bution (bottom right panel). In order to highlight the impactof numerical fragmentation and simplify the comparison ofthe different models to ΛCDM, relative ratios are displayedin the bottom panels and shaded lines represent the distri-bution of the full halo sample, i.e. including also spurioushaloes.

The analytical fit used by Schive et al. (2016) to param-eterize the cumulative HMF drop of the FDM models with

respect to ΛCDM

N(>M)FDM =∫ +∞M

∂MNCDM

[1 +(M

M0

)−1.1]−2.2dM

(21)

with M0 = 1.6× 1010m−4/322 M�, are plotted as reference

– one for each FDM mass – in the top left panel of Fig. 7(dotted lines).

As expected, we find that the number of small masssubhaloes is drastically reduced in the FDM models andthe cumulative distributions depart from ΛCDM at higherand higher masses as the mχ mass decreases. The valuesat which the drop occurs are approximately 5× 1010M�,2.5×1010M� and 5×109M� for values of m22 of 2.5, 5 and25, respectively: this suggest a linear trend of the thresholdmass

Mt ' 5×1010M�( 2.5m22

)(22)

describing the approximate mass below which the numberof haloes starts decreasing with respect to ΛCDM.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8flux

-2.0%

0.0%

2.0%

4.0%

6.0%

P(flu

x)i/P

(flux

) j - 1

[%]


2.5 2.0 1.5 1.0 0.5 0.0 0.5log(1+ )

-100%

0%

100%

200%

300%

400%

500%

P(ga

s)i/P

(gas

) j - 1

[%]


Figure 4. Relative differences of the flux PDF (top panel) andgas PDF (bottom panel) for FDM models with respect to theircorresponding FDMnoQP counterparts, at redshift z = 5.4.

Looking at the distribution of subhaloes masses as com-pared to their associated primary halo M200 and the radialdistribution to R200, it is evident how the numerous smallsubhaloes in ΛCDM, far from the gravitational centre of themain halo, are the ones that were not able to form in a FDMuniverse.

The haloes that have masses above Mt not only havebeen able to survive the disrupting QP action up to red-shift z = 0, but the cumulative distribution shows how theyalso gained extra mass, at the smallest (sub)halo expenses.This is confirmed by the cumulative distribution of the pri-mary structures N(>Mtot−M200), representing the massaccumulated outside the R200 radius, which is systemati-cally higher with respect to ΛCDM case as the FDM masslowers – up to peaks of 200% ratio for the lowest m22 –:this is consistent with the picture of bigger primary haloesaccreting the mass of un-collapsed smaller subhaloes thatdid not form.

The fitting function of Eq. 21 is consistent with the scaleof the drop of the HFM, which is indeed expected to be al-most redshift independent, since it is predominantly givenby the initial PS cut-off (Hu et al. 2000). However, it failsto reproduce the data on two levels: on one hand it does not

recover the slope of the cumulative distribution – especiallyin the mass range close to Mt where the HMF departs fromΛCDM – and, on the other hand, does not account for themass transfer from smaller haloes, unable to collapse dueto QP repulsive interaction, to bigger ones, that accrete themore abundant available matter from their surroundings.The discrepancies between the Schive et al. (2016) fittingfunction and our results are probably due to the fact thatthe former is based on simulations with approximated FDMdynamics and evolved only to redshifts z = 4, thus repre-senting a different collection of haloes that are, moreover, inan earlier stage of evolution.

Therefore, the analysis of the aggregated data of cu-mulative distributions of genuine haloes in each simulationlead us to conclude that formation, the evolution and theproperties of a FDM halo subject to the real effect of theQP – as compared to the FDMnoQP approximation – canfollow three general paths depending on its own mass andon the mass of the FDM boson: if the halo mass isM �Mt,there is high chance that the halo does not form at all sincegravitational collapse is prevented by the QP; if M &Mt,the halo can be massive enough to form but its propertieswill be affected by the QP – especially on its internal struc-ture, as we will see below –, while for M �Mt the halois not severely affected by the QP, and will simply accretemore easily un-collapsed mass available in its surroundings.

In order to study in more detail the impact of FDMon the halo properties and structures, we divided our com-mon sample, that by construction collects the haloes acrossall the simulations that share the same ΛCDM match (asdescribed in detail in Section 3.5), in three contiguous massranges. Let us remind that matching haloes have similar butnot necessarily equal mass, so mass intervals are to be re-ferred to the ΛCDM halo mass; the other matching haloesbelonging to the FDM simulations are free to have lower andhigher mass, compatibly with the limit imposed by the M̃parameter of the common sample selection procedure. Thecommon sample low mass end is clearly limited by the FDM-2.5 model, since it is the one with higher Mt, below whichhaloes have statistically lower chance to form. The threemass ranges are [0.5− 4], [4− 100], [100− 4000]× 1010M�,in order to be compatible with the three halo categories de-scribed in the previous paragraph for the FDM-2.5 model,being Mt(m22 = 2.5)∼ 5×1010M�

For all the matching haloes considered, we have testedthe sphericity distribution, the halo volume and the totalhalo mass with respect to ΛCDM, as well as the radial den-sity profiles.

Properties of inter-simulation matching haloes are gath-ered in Fig. 8, where the total sample is divided column-wisein the three mass ranges. The sphericity, the volume occu-pied and the total mass of the haloes – contrasted with thecorresponding ΛCDM match – are shown in the first row(left panels), together with related distribution functions(right panels). The second and the third row represent theoverall density profiles, stacked in fractional spherical shellsof R200 and ellipsoidal shells of the major axis a – identifiedwith the vertical dashed lines –, respectively, . Density pro-files are divided by the value of the density calculated withinthe R200 and a shells and are shown both in absolute value(top panels) and relatively to ΛCDM (bottom panels).

MNRAS 000, 1–19 (2018)

14 M. Nori et al.

z = 5.4

0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2log10 k [hMpc 1]

-100%

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P(k)

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(k) LC

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1[%

]

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-6%

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]


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]

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-4%

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i/P

(k) j

1[%

]


z = 3.0

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-20%

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0%

5%

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15%

P(k)

i/P

(k) LC

DM

1[%

]

FDM-2.5FDM-5FDM-25

0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2log10 k [hMpc 1]

-2.5%

-2%

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-1%

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0%

0.5%

1%

1.5%

P(k)

i/P

(k) j

1[%

]


Figure 5. Flux power spectrum comparison between all simulations and LCDM (left panels), and between FDM simulation and theirFDMnoQP counterparts (right panels) at different redshifts.

MNRAS 000, 1–19 (2018)


10−3 10−2 10−1

1/mχ [1022/eV]

Pro

bab

ility

dis

trib

uti

on

FDM noQP

FDM

Figure 6. Here we plot the marginalised posterior distributionof 1/mχ from both the analyses performed by Iršič et al. (2017a)(green lines, without QP) and ours (red lines, with QP). Thevertical lines stand for the 2σ C.L. limits.

The sphericity distributions confirm that, in the massrange considered, there is no statistical deviation fromΛCDM, except for a mild deviation towards less sphericalconfigurations of the less massive haloes, especially in them22 = 2.5 model. This is consistent with the analysis of thesphericity distributions of the genuine samples (see lowerpanels in Fig. 1) that reveals that haloes appear to be sta-tistically less spherical with respect to ΛCDM at z = 0 whenlower FDM masses are considered, down to a maximum of∼ 10% decrease in sphericity for m22 = 2.5 and halo mass of∼ 5×109M�.

For all the FDM models the volume occupied by thehaloes is systematically larger, consistently with a delayeddynamical collapse of the haloes. All mass ranges show suchproperty and it is emphasized by lower m22 mass – i.e.stronger QP force –; however, while bigger haloes occupy al-most systematically 20% more volume form22 = 2.5, smallerhaloes can reach even twice the volume occupied by theirΛCDM counterparts when the same model is considered.

Comparing the mass of the haloes in the various modelswith the one in ΛCDM, it is possible to see that small haloesare less massive and big ones, on the contrary, become evenmore massive, confirming our hypothesis of mass transferfrom substructures towards main structures.

The stacked density profiles provide even more insighton the underlying different behaviour between the chosenmass ranges. Starting from the less massive one, the stackedprofiles look very differently if plotted using the sphericalR200-based or the ellipsoidal a-based binning. This is dueto two concurrent reasons related to the properties of thismass range: first of all, as we said before, the sphericityis mχ dependent and thus it is not constant with respectto ΛCDM, so the geometrical difference in the bin shapebecomes important when different models are considered;secondly, since the FDM haloes have lower mass but occupylarger volumes, the two lengths are different from each other– being R200 related to density and a purely to geometry –so that the actual volume sampled is different. Nevertheless,it is possible to see that in FDM models there is an excess ofmass in the outskirts of the halo – seemingly peaking exactlyat distance a – and less mass in the centre.

The intermediate mass range shows also a suppressionin the innermost regions but a less pronounced over-density

around a as expected, since the effectiveness of the repulsiveforce induced by the QP in tilting the density distributiondecreases as its typical scale becomes a smaller fraction ofthe size of the considered objects. In fact, stacked densityprofiles of the most massive haloes are very similar in the twobinning strategies, being R200 ∼ a and sphericity constantamong the various models, and consistent with no majordeviation from ΛCDM, except for a central over-density. Itis our opinion, however, that such feature in the very centreof most massive haloes could be a numerical artefact, sinceits extension is comparable with the spatial resolution used.

The results presented in this Section have been obtainedthrough the detailed analysis of the statistical properties ofhaloes found at z = 0 in the FDM simulations. The sameanalysis, repeated at z = 0, of the FDMnoQP simulationsshows very similar results which are, therefore, not shown inthe present work. Such consistency suggests that the proper-ties of haloes at low redshift are – at the investigated scales –not sensible to modifications induced by the dynamical QPrepulsive effect, which are expected to appear more promi-nently at scales of ∼ 1Kpc with the formation solitonic cores.

5 CONCLUSIONS

We have presented the results obtained from two sets of nu-merical simulations performed with AX-GADGET , an exten-sion of the massively parallel N-body code P-GADGET3 fornon-linear simulations of Fuzzy Dark Matter (FDM) cos-mologies, regarding Lyman-α forest observations and thestatistical detailed characterization of the Large Scale Struc-tures.

More specifically, our main aim was to design a set ofsimulations covering the typical scales and redshifts involvedin Lyman-α forest analyses, in order to extract synthetic ob-servations, compare them with available Lyman-α data, andfinally to place a constraint on the mass of the FDM parti-cle. In the literature, Lyman-α forest was already used forthis purpose but only in approximated set-ups, in which thequantum dynamical evolution of FDM was only encoded inthe initial conditions transfer function, and neglected dur-ing the simulation (Iršič et al. 2017a; Armengaud et al. 2017;Kobayashi et al. 2017), while the AX-GADGET code allowsus to drop such approximation and take into account thenon-linear effects of full FDM dynamics.

The constrain the FDM mass we find is 21.08 ×10−22 eV, which is 3% higher with respect to what was foundin Iršič et al. (2017a). The fact that these two bounds aresimilar, despite the different dynamical evolution consideredin these different works, implies that the additional suppres-sion deriving from the Quantum Potential dynamical con-tribution, at the scales and redshifts probed by Lyman-α ,is compensated by the gravitational growth of perturbationswhen these enter the non-linear regime, implying also that –even if the QP does play a role in the Large Scale Structureevolution – the approximation of Iršič et al. (2017a) (alsoadopted by Armengaud et al. 2017; Kobayashi et al. 2017)is valid and sufficient at these scales.

Secondly, we studied in detail the statistical propertiesof the Large Scale Structures through the analysis of theaggregated data on haloes regarding their mass, volumesand shapes, as well as their individual inner structure.

MNRAS 000, 1–19 (2018)

16 M. Nori et al.

101

102

103

104

Nha

lo(>

Mha

lo)

LCDMSchive16-25FDM-25Schive16-5

FDM-5Schive16-2.5FDM-2.5

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M

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loM

200) LCDMFDM-25

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bhal

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M

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bhal

o(>

R sub

halo

/R20

0) LCDMFDM-25

FDM-5FDM-2.5

10 1 100Rsubhalo/R200

0% 20% 40% 60% 80%100%

N/N

LCD

M

Figure 7. Properties of the halo and subhalo samples at z = 0, with (dashed lines) and without (solid lines) including the haloes markedas spurious as described in Sec. 3.4. In particular, the cumulative distributions of halo mass (top left panel), the halo mass outside R200(top right panel), the subhalo-halo relative mass (bottom left panel) and the subhalo-halo distance (bottom right panel) are displayed.The fitting functions of the cumulative halo mass distribution of (Schive et al. 2016) of Eq. 21 are plotted for reference – dotted line inthe top left panel –.

The main results regarding the effects of FDM on LSSthat we found can be summarized as follows:

• the FDM particle mass m22 defines a typical mass scaleMt ' 1.25×1011/m22 M� characterising the halo distribu-tion of different FDM models; all halo properties can beinterpreted within the framework of having two families ofhaloes: the small ones with M .Mt, and the big ones withM �Mt (since the very small ones M �Mt do not form atall);• small haloes, according to the above definition, show

outward tilted profiles and a lower total mass, are less spher-ical and more voluminous, so less dense overall;• big haloes instead are almost unaffected in their internal

structure – apart from the expected solitonic inner coresthat we cannot resolve with our simulations –, they occupya larger volume and they also have higher total mass, mostlyaccreted outside R200, compatible with the collection of thesubhaloes mass that were not able to form

To conclude, we have performed for the first time a suiteof hydrodynamical simulations of a statistically significantvolume of the universe for Fuzzy Dark Matter models fea-

turing a fully consistent implementation of the QuantumPotential effects on the dynamical evolution of the system.These simulations allowed to perform for the first time afully consistent comparison of mock Lyman-α observationswith available data and to update existing constraints onthe allowed FDM mass range. As the new constraints arenot significantly different from previous ones, this representsthe first direct validation of the approximations adopted inprevious works. Furthermore, our large halo sample allowedus to perform an extensive characterisation of the proper-ties of dark matter haloes in the context of FDM scenarios,highlighting the typical mass scale below which FDM effectsstart to appear. Higher resolution simulations will soon al-low us to explore even smaller scales where we expect toobserve the formation of solitonic cores.

ACKNOWLEDGEMENTS

MN and MB acknowledge support from the Italian Min-istry for Education, University and Research (MIUR)through the SIR individual grant SIMCODE, project num-

MNRAS 000, 1–19 (2018)


[0.5 − 4] × 1010M� [4 − 100] × 1010M� [100 − 4000] × 1010M�

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/20

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Stacked densities of 105 subhaloes, binned in spherical shells

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Figure 8. Properties of inter-simulation matching haloes. The total sample is divided column-wise in three mass ranges. The sphericity,the volume occupied and the total mass of the haloes – contrasted with the corresponding ΛCDM match – are shown in the first row(left panels), together with related distribution functions (right panels). The second and the third row represent the overall densityprofiles, stacked in fractional spherical shells of R200 and ellipsoidal shells of the major axis a – identified with the vertical dashed lines –,respectively. Density profiles are divided by the value of the density calculated within R200 and a and are shown both in absolute value(top panels) and relatively to ΛCDM. (bottom panels)

ber RBSI14P4IH. The simulations described in this workhave been performed on the Marconi supercomputer atCINECA thanks to the PRACE allocation 2016153604. MVand RM are supported by INFN I.S. PD51-INDARK.

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